9702/Oct Nov/11/2011/Q33

Which statement about electrical resistivity is correct?

A The resistivity of a material is numerically equal to the resistance in ohms of a cube of that

material, the cube being of side length one metre and the resistance being measured

between opposite faces.

B The resistivity of a material is numerically equal to the resistance in ohms of a one metre

length of wire of that material, the area of cross-section of the wire being one square

millimetre and the resistance being measured between the ends of the wire.

C The resistivity of a material is proportional to the cross-sectional area of the sample of the

material used in the measurement.

D The resistivity of a material is proportional to the length of the sample of the material used in

the measurement.

Solution:
Answer: A

Let's take this option by option, with the following formula in focus:

ρ = RA/l

Where ρ = resistivity of the material through which current is being passed,
R = resistance of the sample through which current is being passed,
A = Area of the cross section of the sample through which current is being passed (perpendicular to the direction of current), and
l = length of cross section of sample through which current flows (i.e. the distance through which the current flows in the sample used).

So, for option A, let's see what the formula tells us:

If the cube has a one meter side and the current is passed from one face to another, we can say that the current travels 1 meter from one end of the cube to the other; therefore, l = 1 meter. Alternatively, the current first enters the cube on one side, travels 1 meter, and exits the cube, giving us the same value.

Further, the cross section through which the current travels, perpendicular to the direction of current - suppose you place the cube on the table, and cut it parallel to the edges, the area you get after the cut is an area of 1 meter x 1 meter = 1 m^2.
Another way of getting this is to imagine there is no current passing through the cube. You close the circuit, and current starts flowing. Soon, it gets to the beginning of the cube, and starts moving through it. Suppose no charge carrier (electron/proton, either is fine) travels faster than another, you have a "wall" of such charge carriers advancing along the cube. Now ask yourself. How large is that wall? What is the area of that wall? In this case, the area of that wall is 1 m^2, which is our result.

Putting these in the equation, we get

ρ = R * 1 m^2/1 m = R
So ρ = R

Therefore, A is right - the value of ρ is numerically equal to the value of the resistance of the cube in this situation.

Option B: Doing the maths here again, we can say that the length through which the current passes is l = 1 meter and the cross sectional area through which the current passes is 1 mm^2 = (1/1000 meters)^2 = (10^-3)^2 = 10^-6.
Putting it in the equation,

ρ = R * 10^-6/1 = 10^-6 * R
Which is not right.

Option C: the option says that resistivity is dependent on the cross sectional area of the sample used. Even though the formula says that ρ = R/l * A, the wording of the option is something very precise; when it says proportional, what it means is

"...a change in A results in a change in ρ, such that the change in A can be equated to the corresponding change in ρ if a suitable constant k is introduced as a factor of change (i.e. ΔA = k * Δρ)"

This is, of course, false. At a fixed temperature, given no other external electric or magnetic interference (literally and figuratively), the resistivity of a material will not change with any change in area. Instead, the resistance will change to ensure that the value of ρ remains the same. Since resistivity does not depend on the dimensions of a sample and only on the innate nature of the material, C cannot be right.

Changing the cross-sectional area of a sample may change the resistance, but it cannot change the resistivity.

Option D: The same argument as option C can be supplied here, replacing Cross Sectional Area A with length l.

Reference: PYQ - May/Jun 2011 Paper 11 Q33